Several works have focused on the theoretical study of the mass assembly history of fossil and non-fossil systems. One of the first papers on this topic was done by [1]} where they analysed the mass assembly of systems with \(M_{vir} \sim 10^{14}\)  M\(_{\odot }\) . They found a correlation between \(\Delta m_{12}\)  and the formation time of the group defined as the redshift at which 50\(\%\)  of the total mass of the system at \(z = 0\)  is already in place (\(z_{50}\) ). In particular, they found that FGs have assembled more than half of their present mass at \(z > 1\) , with a subsequent growth by minor mergers alone. This early assembly gave to FGs enough time to merge their M\(^{*}\)  galaxies (where M\(^*\)  is the characteristic magnitude of the luminosity function, see Sect. REF  for details) and produce the large magnitude gaps observed at \(z=0\) . The mass assembly of the low-mass group regime (\(M_{vir} \sim 10^{13} - 10^{13.5} M_{\odot }\) ) was analysed by using the Millenium Simulation by [2]}, [3]}. In particular, these authors found that the selection of the systems using their magnitude gap alone does not guarantee the selection of early formed systems. They observed that the majority of the objects that have assembled more than 50\(\%\)  of their halo mass at \(z=1\)  are not fossil systems today. A similar result was found in [4]}: 20\(\%\)  of the groups selected from the Millenium Simulation with large mass gap (similar to fossil systems) turned to be young objects. [5]} proposed that a combination of three observational parameters (magnitude gap, luminosity of the brightest cluster galaxy and its offset from the group luminosity centroid) considerably improve the selected rate of dynamically old systems. In Fig. REF  it can be seen that the probability for a system with \(\Delta m_{12} > 2.0\)  and \(\Delta m_{14} > 2.5\)  to be old grows when the absolute magnitude of the BCG is smaller. In this case, [5]} defined a group as old if its halo has over 50% of its final mass at \(z = 1\)  and young if this fraction is less than 30%. Indeed, this plots demonstrate that \(\Delta m_{14}\)  works better than \(\Delta m_{12}\) , when combined with the absolute magnitude of the central galaxies, in finding old systems. This result was recently confirmed in [7]}, since the authors found a clear correlation between \(\Delta m_{12}\) , the offset of the luminosity centroid, and the dynamical age of a group/cluster. However, this criterium is less used in the literature, probably because it is newer than the [8]} and [3]} ones.
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